Civil Engineering Reference
In-Depth Information
It should be noted that rotation and deflection are functions of moment
only, since shear deformations are ignored in this derivation. Setting
q
iz
to
zero and solving for
P
iy
and
M
iz
in terms of the deflection will result in two
terms of the stiffness matrix.
3
2
∫
Px MP
x
L
23
x
L
dx
EI
q
iz
== −++
0
∆
−
iy
iz
ix
iy
3
2
z
2
L
L
EI
L
EI
0
=−
P
+
M
−
P
∆
iy
iz
ix
iy
2
EI
2
z
z
z
MP
L
1
2
=
+
P
∆
iz
iy
ix
iy
2
2
x
L
4
3
x
L
3
dx
EI
∫
2
∆
=
Px M
−
xxP
+
∆
−
+
iy
iy
iz
ix
iy
3
2
z
3
2
2
L
EI
L
+∆
7
10
L
EI
∆
=
P
−
M
P
iy
iy
iz
ix
iy
3
2
EI
z
z
z
Substituting the first equation into the second equations yields the
following:
3
2
L
EI
L
EI
∆
=
P
+
P
∆
iy
iy
ix
iy
12
10
z
z
12
EI
L
−
6
(5.25)
P
=
∆
z
+
P
∆
iy
iy
ix
iy
3
5
L
Substituting Equation 5.25 into the following equation repeated from ear-
lier results in Equation 5.25 for the second stiffness value.
MP
L
1
2
=
+
P
∆
iz
iy
ix
iy
2
6
EI
L
1
10
−
M
=
∆
z
+
P
∆
(5.26)
iz
iy
ix
iy
2
Take note that the first terms in each of these stiffness equations are the
same as the elastic stiffness values derived in Equations 4.26 and 4.27.
The second term is the geometric component due to the deflected shape
and the axial thrust.
